You seem to be implying a false choice here. Specifically that we (the government is comprised of individuals) can not engage both in labor protections at the same time as we attempt to create a universal health care system.
What is the purpose of the labor protection in the first place ? To help the worker right ? Then why not just directly help the worker.
Instead of forcing company to provide health care to their worker, why not the government itself provide health care directly.
Just because the fix is easier and more likely to achieve but if it doesn't address the core issue, its useless.
This shows up anytime people try to make some sort of progressive improvement to society, people come out of the woodwork to complain that it's not the right solution, without doing anything (other than complaining I guess) to motivate a better solution.
Still better than not saying anything.